I've seen a lot of articles saying that Casper/PoS will be substantially better in terms of energy efficiency but haven't seen any concrete numbers. How much better will it be? 50x (still really bad), 500x (still pretty bad), 5000x (pretty good).

The reason I ask is I'm playing with a new concept I'm calling EfficientChain: Blockchain For Energy Efficiency. Instead of doing useless proof of work calculations, I'm wondering if it's possible to prove consensus through useful energy usage prediction algorithms.

The company I work for has got a relatively rudimentary algo that can predict energy consumption for a given time period based on historical consumption data + meteorology data. I was thinking we could prove consensus based on multiple nodes getting a similar result for the energy usage prediction. (Transaction being a reduction in energy usage).

If proof of stake really will involve negligible energy use (e.g. 10 watt hours per transaction), then maybe we'll scrap that idea and still use Ethereum (plus maybe Golem for doing the calculations). But if PoS is still pretty bad, we'll have to look at alternatives.

1 Answer
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There are no precise details for Casper. However, we can make some assumptions and do some basic math. Firstly, we assume that the proof-of-work part disappears. Proving stake itself is easy. In the worst case, the protocol is communication and regular CPU-heavy. So let's assume we need a high-end computer can stake (it probably wouldn't be) and that there are 100 stakers required. We will ignore sharding; that allows for roughly linear scaling in computation.

Next, we assume that, worst case, the same number of transactions that are processable now, per unit time, are the same as will be processable with PoS. Next, we assume the composition of transactions remains the same with PoS as now. Ethereum has clocked around 1.1M transactions per day -- or ~45k transactions per hour. A slight raising of the gas limit can easily make that 50k.

Now, we assume that our 100 high-end staking computers use 250W each (I'm being overly pessimistic -- that's overkill for a 125W TDP CPU, mobo, fans, with a few SSDs, a bad PSU, and some basic network infrastructure).
That's 25kW for all those stalkers. We'll even make them redundant. That's 50kW (now you see why I rounded earlier)! But at 50k transactions per hour, that's 1Wh per transaction in this scenario I've constructed that is extremely unfavourable towards staking.

If I had to guess, when PoS is first fully in effect and PoW is abolished, the true per-transaction energy use would be closer to 1mWh than to 1Wh (and will continue to drop), assuming the demand for transactions continues to grow (if it doesn't, those staking machines will sit around confirming very few transactions).

Side note:
The PoW calculations are not useless -- they're used to enforce a cost on mining. You can't make work out of thin air. The harder the PoW, the harder it is to take over the network. Also, PoW is not used to prove consensus. It is used to agree on a version of history -- that is, it is used to arrive at consensus. Another purpose for PoW to exist!